Communications in Information and Systems

Volume 21 (2021)

Number 2

Solution existence and uniqueness for degenerate SDEs with application to Schrödinger-equation representations

Pages: 297 – 315

DOI: https://dx.doi.org/10.4310/CIS.2021.v21.n2.a6

Authors

Peter M. Dower (Department of Electrical and Electronic Engineering, University of Melbourne, Victoria, Australia)

Hidehiro Kaise (Department of Mathematics, Kumamoto University, Kumamoto, Japan)

William M. McEneaney (Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, Calif., U.S.A.)

Tao Wang (Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, Calif., U.S.A.)

Ruobing Zhao (Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, Calif., U.S.A.)

Abstract

Existence and uniqueness results for solutions of stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of strong solutions is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solutions of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to this degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold. The second consists of singularities that may occur on a smoothly parameterized curve.

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The research of Hidehiro Kaise was supported in part by JSPS KAKENHI Grant Number 20K03733.

The research of William M. McEneaney and Ruobing Zhao was supported in part by AFOSR Grant FA9550-18-1-0268 and NSF Grant DMS 1908918.

Published 3 June 2021